1. Field of the Invention
The present invention relates generally to the field of linear dynamical systems and particularly to a method and apparatus for decoupling linear dynamical systems.
2. Description of the Prior Art
A linear dynamical system is a certain class of mathematical models of a physical structure, such as for example and without limitation, a building, a bridge, a car, a ship, satellite, airplane or electric circuit. In general, a physical structure, mechanical or electrical, whose dynamic behavior can be predicted with accuracy by a set of linear second-order ordinary differential equations with constant coefficients is, for the purpose of modeling, a linear dynamical system. If the rate at which energy is dissipated within the structure is negligible, the linear dynamical system is termed undamped. If energy dissipation cannot be neglected, the system is said to be damped. In a typical application, the model, i.e. the linear dynamical system, is used to predict the displacement of various components of the physical structure from their equilibrium positions.
A linear dynamical system is generally represented by a set of coupled equations, which are interlinked and cannot be treated independently of each other. This property of coupling is a major barrier to analysis and design. For example, the solution of one equation requires solution of all equations. If coupling can be removed, a set of mutually independent sub-systems, each governed by an independent equation, is obtained. Decoupling is a process by which a complex dynamical system, as represented by a set of coupled equations, is converted into independent sub-systems, with each sub-system specified by only one equation. The significance and advantages of decoupling in analysis and design have long been recognized. In the publication entitled “The Theory of Sound” originally published in 1894 by Lord Rayleigh and later reprinted in 1945 by Dover, N.Y., the well known scientist Lord Rayleigh already expounded on the significance of decoupling and introduced the assumption of proportional damping to decouple a subset of linear dynamical systems. Indeed, the problem of decoupling has attracted the attention of many engineers and scientists in the past century. However, a general procedure for decoupling linear dynamical systems has not been reported.
A time-honored procedure, termed modal analysis or classical modal analysis, known to those of ordinary skill in the art, is available to decouple linear dynamical systems in the absence of damping. An undamped linear system possesses classical normal modes, and in each mode different parts of the system vibrate in a synchronous manner. The normal modes constitute a linear coordinate transformation that decouples the undamped system. This is the essence of modal analysis. In the presence of damping, a linear system cannot be decoupled by modal analysis or by any other prior art technique unless certain restrictive conditions apply. A damped linear system is termed classically damped if it still can be decoupled by modal analysis. Practically speaking, classical damping means that energy dissipation is uniformly distributed throughout the structure. This assumption is violated for systems consisting of two or more parts with significantly different levels of damping. Examples include soil-structure systems, base-isolated structures, and systems in which coupled vibrations of structures and fluids occur. Increasing use of special energy-dissipating devices in control constitutes another important class of examples. In fact, experimental modal testing suggests that no mechanical system is classically damped. Thus, modal analysis is generally not applicable to damped linear systems.
Due to the lack of an exact method for decoupling, engineers routinely invoke a whole array of approximations to continue to base the analysis of linear dynamical systems upon modal analysis. From a practical viewpoint, there is no objection to using approximate techniques to decouple a linear dynamical system. However, rigorous analysis of the errors of approximation committed by these mostly ad hoc techniques has not been conducted. In general, every approximate technique is only suitable for decoupling a subset of linear dynamical systems satisfying certain restrictive assumptions; it is otherwise inadequate for other applications. There is not any prior art technique, either approximate or exact, that is adequate for decoupling any linear dynamical system.
An alternative but more abstract approach to linear dynamical systems is the state space approach, known to those of ordinary skill in the art. In this approach, the second-order equations governing a linear system are recast in a first-order form known as state equations. The problem with this approach is that the number of equations describing the system is artificially doubled. The state space approach has not appealed to practicing engineers, particularly in the fields of structural mechanics and materials. The inordinate amount of computational effort due to the doubling of the number of equations is usually given as a reason. More importantly, there is serious loss of physical insight in tackling the first-order state equations. In addition, unless it is assumed that the state companion matrix is diagonalizable, the state equations still cannot be decoupled exactly with prior art techniques. In short, decoupling in state space analysis is inadequate in many ways.
For the foregoing reasons, there is a need for a method and apparatus for exact decoupling of linear dynamical systems as represented by a set of coupled second-order ordinary differential equations. There is further a need for such a decoupling methodology to be amenable to physical interpretation so as to facilitate and streamline the analysis and design of linear dynamical systems.